Interpretation of a parameter table of NLME analysis

Shen Cheng

2026-01-28

NLME parameter table (fixed-effect parameters)1

NLME parameter table example (random-effect parameters)1

Fixed-effect parameters

  • Parameter descriptions

    • Parameter abbreviations with unit: CL/F (L/h)

    • Parameter labels (optional): \(\theta_1\)

    • Parameter descriptions: Apparent central clearance

  • Footnotes:

    • Full name for abbreviations?
    • How were RSE or CI calculated

  • Parameter values:

    • Point estimates

      • Interpretatability: For example, if parameterize \(log(CL)=THETA(1)\) during modeling, reporting \(exp(THETA(1))\) for CL in normal scale.

    • Uncertainty (i.e., precisions or imprecisions)

      • Standard error (SE)

      • Relative standard error (RSE)

      • 95% confidence interval (CI)

Fixed-effect parameters: Point estimates

  • Output file:.ext
  • Final point estimates in the row with Iteration=-1000000000.

Point estimates

Back-transform model parameters when necessary

  • For example, if a parameter is modeled in:

    • log scale (log-transformation):

      • NONMEM: \(CL=EXP(THETA(1))\)

      • Back-transform: report \(e^{\theta_1}\) instead of \(\theta_1\).

  • logit scale (logit-transformtion):

    • NONMEM: \(F=EXP(THETA(1))/(EXP(THETA(1)+1))\)

    • Back-transform: report \(\frac{e^{\theta_1}}{e^{\theta_1}+1}\) instead of \(\theta_1\).

Fixed-effect parameters: Uncertainty (SE)

  • Output file: .cov
    • SE of a parameter: \(SE=sqrt{diag}\)
    • Although also available in .ext row Iteration=-10000000001.

\[ \begin{bmatrix} \mathbf{x_{11}} & x_{12} & x_{13} & \dots & x_{1n} \\ x_{21} & \mathbf{x_{22}} & x_{23} & \dots & x_{2n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ x_{d1} & x_{d2} & x_{d3} & \dots & \mathbf{x_{dn}} \end{bmatrix} \]

Fixed-effect parameters: Uncertainty (RSE)

If a parameter is in

  • Normal scale: \(RSE=\frac{SE}{\hat{\theta}} \times 100\%\)

  • Log scale: \(RSE=sqrt{e^{diag}-1}\)

    • \(diag\): a diagonal element of the covariance matrix (i.e., variance of SE).

If a parameter is in

  • Logit scale:

    • \(RSE=\frac{SE}{\hat{\theta}} \times 100\%\) (may not appropriate)

    • \(RSE=\frac{SE^{*}}{\hat{\theta^{*}}} \times 100\%\)

      • \(\hat{\theta^{*}} = \frac{e^{\hat{\theta}}}{e^{\hat{\theta}}+1}\)

      • \(SE^{*}=SE \times \theta\times (1-\theta)\) (Delta method)

Fixed-effect parameters: Uncertainty (confidence interval)

  • Compute CI of transformed parameters:
    • Parametric (symmetrical):
      • Assuming normal distributions: estimates \(\pm\) 1.96*SE.
    • Non-parametric (asymmetrical):
      • Bootstrap, LLP, SIR and Bayesian (week 6).

Random-effect parameters

  • Majority of the elements are similar to the fixed-effect parameters:
    • Parameter descriptions, abbreviations, and labels.
    • Point estimates (Iteration=-1000000000 in .ext).
    • Uncertainty: SE, RSE or CV.
    • Footnotes:
      • Full name for abbreviations?
      • How were RSE or CI calculated
  • Report \(CV\%\) for BSV terms.
  • Report \(Corr\) for covariance terms.
  • Report \(CV\%\) or \(SD\) for proportional or additive RUV, respectively
  • Report shrinkage.
  • Footnotes: how were \(CV\%\) and \(SD\)of BSV and RUV calculated.

Random-effect parameters: CV% of BSV

  • A log-normal distribution is commonly used for model BSV.
    • Avoid negative values in individual PK parameters (e.g., CL).
    • CL=TVCL* EXP(ETA(1))
    • \(CV\% = sqrt{e^{\omega_{1,1}^2}-1} \times 100\%\)
  • If a normal distribution is used instead
    • CL=TVCL+ETA(1)
    • \(CV\%=sqrt{\omega_{1,1}^2} \times 100\%\)
  • If a logit-normal distribution is used:
    • F=EXP(THETA(1)+ETA(1))/(EXP(THETA(1)+ETA(1))+1)
    • Variability depends on mean.
    • \(CV\%\) is not appropriate, \(SD\) is calculated instead using R functions like logitnorm::momentsLogitnorm1

Random-effect parameters: Corr of covariance terms

  • \(Corr=\frac{COV_{p1,p2}}{sqrt{VAR_{p1}} \times sqrt{VAR_{p2}}}\)
    • \(COV_{p1,p2}\): estimated covariance between parameters 1 and 2 (p1 and p2).
    • \(VAR_{p1}\): estimated variance of p1.
    • \(VAR_{p2}\): estimated variance of p2.
  • Example: \(Corr_{(CL/F,V2/F)}=\frac{0.0703}{sqrt{0.114} \times sqrt{0.0824}}=0.0725\)

Random-effect parameters: CV% or SD for RUV

Combined error model: Y = IPRED*(1+EPS(1))+EPS(2)

  • Report \(CV\%\) for proportional error term: \(CV\%=sqrt{\sigma^2_{1,1}}\)
  • Report \(SD\) for additive error term: \(SD=sqrt{\sigma^2_{2,2}}\)

Random-effect parameters: Shrinkage (\(\eta\) shrinkage)

  • When the individual data brings only few information about the individual parameter value, the individual estimates (e.g., empirical Bayes estimates, EBEs) is close to (or “shrinks” to) population mean1.
  • Output file:.shk
  • \(\eta_i-shrinkage = 1-\frac{SD_{\eta_i}}{\omega}\) (standard)
  • \(\eta_i-shrinkage = 1-\frac{VAR_{\eta_i}}{\omega^2}\)

\(\eta\) shrinkage1

  • High shrinkage (e.g., SD shrinkage > 30%):
    • Doesn’t necessarily mean the model is unacceptable.
    • Typically occurs when individual data is sparse (e.g., sparse PK sampling).
      • So inadequate information to estimate the EBE of a parameter.
      • EBE distribution may not reflect the actual ones.
    • Diagnostic plots using EBEs are misleading (hiding or suggesting wrong ones):
      • Correlation among \(\eta\)s.
      • Covariate-parameter relationship.
      • DV vs. IPRED

\(\eta\) shrinkage1

Random-effect parameters: Uncertainty (confidence interval)

  • Compute CI of transformed parameters:
    • Parametric (symmetrical):
      • Assuming normal distributions: estimates \(\pm\) 1.96*SE.
    • Non-parametric (asymmetrical):
      • Bootstrap, LLP, SIR and Bayesian (week 6).